The compatibility "demonstrated" by Rovelli & al. between various "gauge
conditions" both in Classical Electromagnetism and General Relativity can be
better understood if one distinguishes "gauge conditions" of the solution type
and "gauge conditions" of the constraint type

Rousseaux, Germain

English

arXiv

The compatibility "demonstrated" by Rovelli & al. between various "gauge conditions" both in Classical Electromagnetism and General Relativity can be better understood if one distinguishes "gauge conditions" of the solution type and "gauge conditions" of the constraint type

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Comment on the paper by Rovelli & al. about the
compatibility of various ”gauge conditions”
Germain Rousseaux
To cite this version:
Germain Rousseaux. Comment on the paper by Rovelli & al. about the compatibility of various
”gauge conditions”. 2007. <hal-00154085>
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Comment on the paper by Rovelli & al. about the compatibility
of various ”gauge conditions”
Germain Rousseaux
Universite´ de Nice Sophia-Antipolis,
Institut Non-Line´aire de Nice,
UMR 6618 CNRS-UNSA,
1361, route des Lucioles. 06560 Valbonne, France.
(Germain.Rousseaux@inln.cnrs.fr)
The compatibility ”demonstrated” by Rovelli & al. in [1] between various ”gauge conditions” both
in Classical Electromagnetism and General Relativity can be better understood if one distinguishes
”gauge conditions” of the solution type and ”gauge conditions” of the constraint type.
Recently, we showed that the well-known Coulomb and
Lorenz ”gauge conditions” (see [2] for an introduction to
the various ”gauge conditions” encountered in the liter-
ature) were, in fact, not equivalent because they must
be interpreted as physical constraints that is electromag-
netic continuity equations [3]. In addition, we were able
to demonstrate that the Coulomb ”gauge condition” is
the galilean approximation of the Lorenz ”gauge condi-
tion” within the magnetic limit of Le´vy-Leblond & Le
Bellac [4, 5, 6, 7]. So, to ”make a gauge choice” that is
choosing a gauge condition is, as a consequence of our
findings, not related to the fact of fixing a special couple
of potentials. Gauge conditions are completely uncorre-
lated to the supposed indeterminacy of the potentials.
Hence, we propose to rename the ”gauge conditions” like
the ones of Lorenz or Coulomb by physical ”constraints”.
The purpose of our comment is to show that a
”demonstration” of the compatibility between the Fock-
Schwinger ”gauge condition” and the Lorenz ”gauge con-
ditions” by Rovelli & al. in [1] is blurred by a slip in
terminology. As a matter of fact, the Lorenz equation
∂µA
µ = 0 is according to us a constraint whereas the
Fock-Schwinger equation xµA
µ = 0 is what we called a
”solution” submitted to the Lorenz constraint. If so, the
”compatibility” is obvious...
First of all, let us illustrate what we mean by a ”so-
lution” under a physical constraint. One often finds in
textbooks that we can describe a uniform magnetic field
B = Bez by either the so-called symmetric ”gauge”
A1 = 1/2B× r or by the so-called Landau ”gauge” [2].
This two ”gauges” are related by a gauge transformation :
A1 =
1
2
B× r =
1
2
[−By,Bx, 0] (1)
becomes either :
A2 = [0, Bx, 0] or A3 = [−By, 0, 0] (2)
with the gauge functions ±f = ±xy/2.
However, there is no discussion in the litterature of
the following issue. As a matter of fact, if we consider
a solenoid with a current along eθ, the magnetic field is
uniform (along ez) and could be described by the sym-
metric ”gauge” or the Landau ”gauge”. Yet, the vector
potential in the Landau ”gauge” A2 is along ey whereas
the vector potential in the symmetric gauge is along eθ.
We advocate that only the symmetric ”gauge” is valid
in this case because it does respect the symmetry of the
currents (J = Jeθ) whereas the Landau ”gauge” does
not. Moreover, the symmetric ”gauge” (or the Landau
”gauge”) is not, in fact, a gauge condition but a solution
describing a uniform magnetic field under the Coulomb
constraint (∇.A1 = 0). In order to understand this last
point, one can picture an analogy between Fluid Mechan-
ics and Classical Electromagnetism. Indeed, the solenoid
is analogous to a cylindrical vortex core with vorticity w
and we know that the velocity inside the core is given by
u = 1/2w×r which is analogous to the symmetric gauge
for an incompressible flow (∇.u = 0). Outside the vortex
core, the velocity is given by [8] :
u =
Γ∇θ
2pi
=
Γ
2pir
eθ (3)
where Γ is the flux of vorticity inside the vortex or the
circulation of the velocity outside the vortex. One recov-
ers the analogue formula for the vector potential outside
a solenoid...
Of course, if the problem we are considering does not
feature the cylindrical geometry (two horizontal plates
with opposite surface currents for example, analogous to
a plane Couette flow [8]), one of the Landau ”gauges”
A2 or A3 submitted to the constraints ∇.A2 = 0
or ∇.A3 = 0 must be used instead of the symmetric
”gauge” A1 according to the necessity of respecting the
underlying distribution/symmetry of the currents which
is at the origin of both the vector potential and the mag-
netic field. To give a magnetic vector field without speci-
fying its current source is an ill-posed problem which was
interpreted so far by attributing an indeterminacy to the
vector potential which is wrong. Now, how can we test
experimentally this argument based on symmetry ? If the
current of the solenoid varies with time, it will create an
electric field which is along eθ as the vector potential be-
cause the electric field is minus the time derivative of the
2vector potential. If the currents in the horizontal plates
change with time, a horizontal electric field will appear
for the same reason.
We come back now to the main claim of Rovelli & al.
[1]. We will argue that the Fock-Schwinger equation is a
solution submitted to the Lorenz constraint. Let us write
the Fock-Schwinger ”gauge” xµA
µ = 0 in projection for
cartesian coordinates x0A
0 + x1A
1 + x2A
2 + x3A
3 = 0.
The four-position and four-potential have components
xµ = (−ct,x) and A
µ = (V/c,A). For example, the Fock-
Schwinger equation becomes xAx + yAy + zAz = tV in
cartesian coordinates.
The Lorenz ”gauge” ∇.A + 1/c2∂V/∂t = 0 becomes
after a Fourier transformation ik.A− iω/c2V = 0. But a
light wave is such that |k| = ω/c.
A plane or spherical light wave has a null space-time
interval (ds2 = dxµ.dx
µ = 0) : x2 + y2 + z2 = c2t2 or
r2 = c2t2. The Lorenz constraint for a plane or spherical
light wave becomes : V = cAx or V = cAr where Ax or
Ar is the component of the light wave which is parallel
to the wave-vector k. With x = ct (y = z = 0) or r =
ct for outgoing waves, one gets V = cAx = x/tAx or
V = cAr = r/tAr that is the Fock-Schwinger solution :
xAx = tV or rAr = tV .
As a conclusion, the Fock-Schwinger solution describes
the propagation of a light wave submitted to the Lorentz-
covariant Lorenz constraint which is the continuity equa-
tion allowing the propagation. It is interesting to notice
that V. A. Fock himself derived the Fock-Schwinger equa-
tion starting with the Lorenz equation and using a gauge
transformation [9]...
[1] E. Magliaro, C. Perini & C. Rovelli, Compatibility of ra-
dial, Lorenz and harmonic gauges, preprint, May 2007.
(Article available online at :
http : //fr.arxiv.org/abs/0704.0992v2)
[2] J. D. Jackson & L. B. Okun, Historical roots of gauge
invariance, Reviews of Modern Physics, Vol. 73, p. 663-
680, 2001.
[3] G. Rousseaux, On the physical meaning of the gauge con-
ditions of Classical Electromagnetism : the hydrodynam-
ics analogue viewpoint, Annales de la Fondation Louis de
Broglie, Volume 28, Nume´ro 2, p. 261-270, 2003.
(Article available online at :
http : //www.ensmp.fr/aflb/AFLB − 282/
aflb282p261.pdf)
[4] M. Le Bellac & J.-M. Le´vy-Leblond, Galilean Electromag-
netism, Il Nuovo Cimento, Vol. 14B, N. 2, 11 Aprile, p.
217-233, 1973.
[5] G. Rousseaux, Lorenz or Coulomb in Galilean Electro-
magnetism ?, EuroPhysics Letters, 71 (1), p. 15-20, July
2005.
[6] M. de Montigny & G. Rousseaux, On the electrodynamics
of moving bodies at low velocities, European Journal of
Physics, Volume 27, Number 4, p. 755-768, July 2006.
[7] M. de Montigny & G. Rousseaux, On some applications
of the galilean electrodynamics of moving bodies, to be
published, 2007.
[8] E. Guyon, J.-P. Hulin, L. Petit L. & C. D. Mitescu, Phys-
ical Hydrodynamics, Oxford University Press, 2001.
[9] V. A. Fock, Proper Time in Classical and Quantum
Mechanics in Selected Works, Edited by L. D. Fadeev,
L. A. Khalfin, I. V. Komarov, p. 421-439, Chapman &
Hall/CRC, 2004.


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